You can use GridMaster to construct electronic multigrid paper. Start by moving the green and blue vectors -- click on a circle at the end of a vector and drag it to a new position. Then click on the appropriate button to display a grid, e.g., a green grid corresponding to the green vectors. You can resize the scale of the entire grid space by moving the point e(1) if you need more a larger or smaller range of values. In the upper left, you will see the coordinates of the point P (the red vector) with respect to the natural basis and then with respect to the bases [P]B defined by the blue vectors and [P]Gdefined by the green vectors.

Note: The "?" at the bottom right-hand corner of the workspace is a link to Key Curriculum Press and its About JavaSketchpad web page.

Sample Exploratory Activity:

Find the vector P determined by the given coordinate vector [P](u,v) and the given basis {u,v}.

,

,

Given the vector P and a basis {u,v}, find the coordinate vector [P](u,v).

,

,

Use the information provided to determine the missing element.

, , and . Find .

, , and . Without attempting to coordinatize P with respect to the standard basis, find a basis G that satisfies these conditions, if any exists.

This last question begins to motivate a need to construct a change-of-coordinates matrix in order to determine more readily if a solution exists. Although there are an infinite number of solutions, determining a single one using GridMaster requires careful attention to the interaction of vectors and how they impact the movement of the grid lines and values of the coordinatized vectors.